On cyclicity in de Branges-Rovnyak spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929757956866048 |
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| author | Bergman, Alex |
| author_facet | Bergman, Alex |
| contents | We study the problem of characterizing the cyclic vectors in de Branges-Rovnyak spaces. Based on a description of the invariant subspaces we show that the difficulty lies entirely in understanding the subspace $(aH^{2})^{\perp}$ and give a complete function theoretic description of the cyclic vectors in the case $\dim (aH^{2})^{\perp} < \infty$. Incidentally, this implies analogous results for certain generalized Dirichlet spaces $\mathcal{D}(μ)$. Most of our attention is directed to the infinite case where we relate the cyclicity problem to describing the exposed points of $H^{1}$ and provide several sufficient conditions. A necessary condition based on the Aleksandrov-Clark measures of $b$ is also presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_09601 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On cyclicity in de Branges-Rovnyak spaces Bergman, Alex Functional Analysis Complex Variables 46E22 We study the problem of characterizing the cyclic vectors in de Branges-Rovnyak spaces. Based on a description of the invariant subspaces we show that the difficulty lies entirely in understanding the subspace $(aH^{2})^{\perp}$ and give a complete function theoretic description of the cyclic vectors in the case $\dim (aH^{2})^{\perp} < \infty$. Incidentally, this implies analogous results for certain generalized Dirichlet spaces $\mathcal{D}(μ)$. Most of our attention is directed to the infinite case where we relate the cyclicity problem to describing the exposed points of $H^{1}$ and provide several sufficient conditions. A necessary condition based on the Aleksandrov-Clark measures of $b$ is also presented. |
| title | On cyclicity in de Branges-Rovnyak spaces |
| topic | Functional Analysis Complex Variables 46E22 |
| url | https://arxiv.org/abs/2309.09601 |